Question

You buy a bucket filled with white and yellow ping-pong balls, covered with a lid. The seller claims that at least half the balls in the bucket are yellow.

The seller offers you a test as evidence. You get to draw 8 ping-pong balls from the bucket with a replacement but without peeking. Assume that each ball is equally likely to be drawn.

a) What is your null hypothesis about the probability (p) to draw a yellow ball?

b) If all 8 ping-pong balls turn out to be yellow, what is the p-value in your test?

c) Is this evidence strong enough to reject the null hypothesis and prove the seller’s claim?

d) What if 6 of 8 ping-pong balls turn out to be yellow - what is the p-value now, and is this evidence still strong enough?

e) What is the smallest number of yellow balls we could draw and still get a p-value under 5%?

Answer 1

(a) Here null hypothesis : There is less  than 50% balls in the bucket are yellow. p < 0.50

Alternative Hypothesis : There is more than 50% balls in the bucked are yellow. p > 0.50

(b) Here probability of having a yellow ball, when n = 8 and p = 0.5

x ~ BINOMIAL (n = 8; p = 0.5)

p - value = P(x = 8; n = 8; p = 0.5) = 8C8 (0.5)8 = 0.0039

(c) Yes this evidence is strong enought to reject the null hypothesis and prve the seller's claim.

(d) Now here x = 6

p - value = P(x 6; 8 ; 0.5) = P(x = 6) + P(X =7) + P(X = 8)

= 8C6 (0.5)8 + 8C7 (0.5)8 + 8C8 (0.5)8 = 0.1094 + 0.03125 + 0.0039 = 0.1445

No here the evidence is not strong as p - value > 0.05.

(e) As we can see from above

P(x =7) + P(x = 8) = 0.0352 < 0.05

so here  smallest number of yellow balls we could draw and still get a p-value under 5% is 7(Seven).